Octal
Numeral systems by culture 


Positional systems by base 
Nonstandard positional numeral systems 
List of numeral systems 
The octal numeral system, or oct for short, is the base8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping consecutive binary digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010, which can be grouped into (00)1 001 010 – so the octal representation is 112.
In the decimal system each decimal place is a power of ten. For example:
In the octal system each place is a power of eight. For example:
By performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal.
×  1  2  3  4  5  6  7  10 
1  1  2  3  4  5  6  7  10 
2  2  4  6  10  12  14  16  20 
3  3  6  11  14  17  22  25  30 
4  4  10  14  20  24  30  34  40 
5  5  12  17  24  31  36  43  50 
6  6  14  22  30  36  44  52  60 
7  7  16  25  34  43  52  61  70 
10  10  20  30  40  50  60  70  100 
Usage
By Native Americans
The Yuki language in California and the Pamean languages^{1} in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.^{2}
By Europeans
 It has been suggested that the reconstructed ProtoIndoEuropean word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that protoIndoEuropeans used an octal number system, though the evidence supporting this is slim.^{3}
 In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number system based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as the base. In 1718 Swedenborg wrote (but did not publish) a manuscript: "En ny räknekonst som omväxlas vid talet 8 istället för det vanliga vid talet 10" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 17 are there denoted by the consonants l, s, n, m, t, f, u (v) and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule.^{4}
 Writing under the pseudonym "Hirossa ApIccim" in The Gentleman's Magazine, (London) July 1745, Hugh Jones proposed an octal system for British coins, weights and measures. "Whereas reason and convenience indicate to us an uniform standard for all quantities; which I shall call the Georigan standard; and that is only to divide every integer in each species into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho' all nations count universally by tens (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ten is uncapable...." In a later treatise on Octave computation (1753) Jones concluded: "Arithmetic by Octaves seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic."^{5}
 In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic. He suggested base 8 for which he coined the term octal. His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of English units was already, to a remarkable extent, an octal system.^{6}
 In the mid 19th century, Alfred B. Taylor concluded that "Our octonary [base 8] radix is, therefore, beyond all comparison the "best possible one" for an arithmetical system." The proposal included a graphical notation for the digits and new names for the numbers, suggesting that we should count "un, du, the, fo, pa, se, ki, unty, untyun, untydu" and so on, with successive multiples of eight named "unty, duty, thety, foty, paty, sety, kity and under." So, for example, the number 65 would be spoken in octonary as underun.^{7}^{8} Taylor also republished some of Swedenborg's work on octonary as an appendix to the abovecited publications.
In fiction
In the 2009 film Avatar, the language of the extraterrestrial Na'vi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand.^{9}
In the TV series Stargate SG1, the Ancients, a race of beings responsible for the invention of the Stargates, used an octal system of mathematics.
In the tabletop game series Warhammer 40,000, the Tau race use an octal number system.
In computers
Octal became widely used in computing when systems such as the PDP8, ICL 1900 and IBM mainframes employed 12bit, 24bit or 36bit words. Octal was an ideal abbreviation of binary for these machines because their word size is divisible by three (each octal digit represents three binary digits). So four, eight or twelve digits could concisely display an entire machine word. It also cut costs by allowing Nixie tubes, sevensegment displays, and calculators to be used for the operator consoles, where binary displays were too complex to use, decimal displays needed complex hardware to convert radices, and hexadecimal displays needed to display more numerals.
All modern computing platforms, however, use 16, 32, or 64bit words, further divided into eightbit bytes. On such systems three octal digits per byte would be required, with the most significant octal digit representing two binary digits (plus one bit of the next significant byte, if any). Octal representation of a 16bit word requires 6 digits, but the most significant octal digit represents (quite inelegantly) only one bit (0 or 1). This representation offers no way to easily read the most significant byte, because it's smeared over four octal digits. Therefore, hexadecimal is more commonly used in programming languages today, since two hexadecimal digits exactly specify one byte. Some platforms with a poweroftwo word size still have instruction subwords that are more easily understood if displayed in octal; this includes the PDP11 and Motorola 68000 family. The modernday ubiquitous x86 architecture belongs to this category as well, but octal is rarely used on this platform, although certain properties of the binary encoding of opcodes become more readily apparent when displayed in octal, e.g. the ModRM byte, which is divided into fields of 2, 3, and 3 bits, so octal can be useful in describing these encodings.
Octal is sometimes used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with file permissions under Unix systems (see chmod). It has the advantage of not requiring any extra symbols as digits (the hexadecimal system is base16 and therefore needs six additional symbols beyond 0–9). It is also used for digital displays.
In programming languages, octal literals are typically identified with a variety of prefixes, including the digit 0, the letters o or q, or the digit–letter combination 0o. In Motorola convention, octal numbers are prefixed with @, whereas a small letter o is added as a postfix following the Intel convention. DRDOS DEBUG uses \ to prefix octal numbers.
For example, the literal 73 (base 8) might be represented as 073, o73, q73, 0o73, \73, @73 or 73o in various languages.
Newer languages have been abandoning the prefix 0, as decimal numbers are often represented with leading zeroes. The prefix q was introduced to avoid the prefix o being mistaken for a zero, while the prefix 0o was introduced to avoid starting a numerical literal with an alphabetic character (like o or q), since these might cause the literal to be confused with a variable name. The prefix 0o also follows the model set by the prefix 0x used for hexadecimal literals in the C language; it is supported by Haskell,^{10} OCaml,^{11} Perl 6,^{12} Python as of version 3.0,^{13} Ruby,^{14} Tcl as of version 9,^{15} and it is intended to be supported by ECMAScript 6^{16} (the prefix 0 has been discouraged in ECMAScript 3 and dropped in ECMAScript 5^{17}).
Octal numbers that are used in some programming languages (C, Perl, PostScript…) for textual/graphical representations of byte strings when some byte values (unrepresented in a code page, nongraphical, having special meaning in current context or otherwise undesired) have to be to escaped as \nnn. Octal representation of nonASCII bytes may be particularly handy with UTF8, where any start byte has octal value \3nn and any continuation byte has octal value \2nn.
Conversion between bases
Decimal to octal conversion
Method of successive division by 8
To convert integer decimals to octal, divide the original number by the largest possible power of 8 and successively divide the remainders by successively smaller powers of 8 until the power is 1. The octal representation is formed by the quotients, written in the order generated by the algorithm. For example, to convert 125_{10} to octal:
 125 / 8^{2} = 1
 125 − 8^{2} × 1 = 61
 61 / 8^{1} = 7
 61 − 8^{1} × 7 = 5
 5 / 8^{0} = 5
Therefore, 125_{10} = 175_{8}.
Another example:
 900 / 8^{3} = 1
 900 − 8^{3} × 1 = 388
 388 / 8^{2} = 6
 388 − 8^{2} × 6 = 4
 4 / 8^{1} = 0
 4 − 8^{1} × 0 = 4
 4 / 8^{0} = 4
Therefore, 900_{10} = 1604_{8}.
Method of successive multiplication by 8
To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds.
Example: Convert 0.1640625 to octal:
 0.1640625 × 8 = 1.3125 = 1 + 0.3125
 0.3125 × 8 = 2.5 = 2 + 0.5
 0.5 × 8 = 4.0 = 4 + 0
Therefore, 0.1640625_{10} = 0.124_{8}.
These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part.
Octal to decimal conversion
To convert a number k to decimal, use the formula that defines its base8 representation:
In this formula, a_{i} is an individual octal digit being converted, where i is the position of the digit (counting from 0 for the rightmost digit).
Example: Convert 764_{8} to decimal:
 764_{8} = 7 × 8^{2} + 6 × 8^{1} + 4 × 8^{0} = 448 + 48 + 4 = 500_{10}
For doubledigit octal numbers this method amounts to multiplying the lead digit by 8 and adding the second digit to get the total.
Example: 65_{8} = 6 × 8 + 5 = 53_{10}
Octal to binary conversion
To convert octal to binary, replace each octal digit by its binary representation.
Example: Convert 51_{8} to binary:
 5_{8} = 101_{2}
 1_{8} = 001_{2}
Therefore, 51_{8} = 101 001_{2}.
Binary to octal conversion
The process is the reverse of the previous algorithm. The binary digits are grouped by threes, starting from the least significant bit and proceeding to the left and to the right. Add leading 0s (or trailing zeros to the right of decimal point) to fill out the last group of three if necessary. Then replace each trio with the equivalent octal digit.
For instance, convert binary 1010111100 to octal:

001 010 111 100 1 2 7 4
Therefore, 1010111100_{2} = 1274_{8}.
Convert binary 11100.01001 to octal:

011 100 . 010 010 3 4 . 2 2
Therefore, 11100.01001_{2} = 34.22_{8}.
Octal to hexadecimal conversion
The conversion is made in two steps using binary as an intermediate base. Octal is converted to binary and then binary to hexadecimal, grouping digits by fours, which correspond each to a hexadecimal digit.
For instance, convert octal 1057 to hexadecimal:
 To binary:

1 0 5 7 001 000 101 111
 then to hexadecimal:

0010 0010 1111 2 2 F
Therefore, 1057_{8} = 22F_{16}.
Hexadecimal to octal conversion
Hexadecimal to octal conversion proceeds by first converting the hexadecimal digits to 4bit binary values, then regrouping the binary bits into 3bit octal digits.
For example, to convert 3FA5_{16}:
 To binary:

3 F A 5 0011 1111 1010 0101
 then to octal:

0 011 111 110 100 101 0 3 7 6 4 5
Therefore, 3FA5_{16} = 37645_{8}.
See also
 Computer numbering formats
 Octal games, a game numbering system used in combinatorial game theory
References
 ^ Avelino, Heriberto (2006). "The typology of Pame number systems and the limits of Mesoamerica as a linguistic area". Linguistic Typology 10 (1): 41–60. doi:10.1515/LINGTY.2006.002.
 ^ Marcia Ascher. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal. Retrieved 20070413.
 ^ Winter, Werner (1991). "Some thoughts about IndoEuropean numerals". In Gvozdanović, Jadranka. IndoEuropean numerals. Trends in Linguistics 57. Berlin: Mouton de Gruyter. pp. 13–14. ISBN 3110113228. Retrieved 20130609.
 ^ Donald Knuth, The Art of Computer Programming
 ^ See H.R. Phalen, "Hugh Jones and Octave Computation," The American Mathematical Monthly 56 (August–September 1949): 46165.
 ^ James Anderson, On Octal Arithmetic [title appears only in page headers], Recreations in Agriculture, NaturalHistory, Arts, and Miscellaneous Literature, Vol. IV, No. 6 (Feb. 1801), T. Bensley, London; pages 437448.
 ^ A.B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, Sept. 15, 1859. See pages and 48 and 53.
 ^ Alfred B. Taylor, Octonary numeration and its application to a system of weights and measures, Proc. Amer. Phil. Soc. Vol XXIV, Philadelphia, 1887; pages 296366. See pages 327 and 330.
 ^ Counting in Na'vi
 ^ Haskell: http://www.haskell.org/onlinereport/lexemes.html#sect2.5
 ^ OCaml: http://caml.inria.fr/pub/docs/manualocaml/lex.html
 ^ Perl 6: http://perlcabal.org/syn/S02.html#Radix_markers
 ^ Python 3: http://docs.python.org/3.1/reference/lexical_analysis.html#integerliterals
 ^ RubySpec: https://github.com/kostya/rubyspec/blob/master/core/string/to_i_spec.rb
 ^ Tcl: http://wiki.tcl.tk/498
 ^ ECMAScript 6th Edition draft: https://people.mozilla.org/~jorendorff/es6draft.html#secliteralsnumericliterals
 ^ Mozilla Developer Network: https://developer.mozilla.org/enUS/docs/Web/JavaScript/Reference/Global_Objects/parseInt
External links
 Octomatics is a numeral system enabling simple visual calculation in octal.
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